Maximum Likelihood Estimator

1. The problem statement, all variables and given/known data
1. An experiment consists of giving a sequences of patients a risky treatment until two have died, and then recording N, the number who survived. If p is the proportion killed by the treatment, then the distribution of N is:
P(N=n)=(n+1)(1-p)n p2

1)Find a general formula for the MLE for p:
2)the experiment is done in 8 hospitals and the observed values of N are 3,0,4,2,3,5,1,3. Compute the estimate for the p derived in part 1

2. Relevant equations2. I know I have to start by finding a general form of ∏(ni+1)(1-p)ni p2 with ni where i goes from 1 to m.

3. The attempt at a solution3. So I get hung up on how to handle the (ni+1) term.

1. The problem statement, all variables and given/known data
1. An experiment consists of giving a sequences of patients a risky treatment until two have died, and then recording N, the number who survived. If p is the proportion killed by the treatment, then the distribution of N is:
P(N=n)=(n+1)(1-p)n p2

1)Find a general formula for the MLE for p:
2)the experiment is done in 8 hospitals and the observed values of N are 3,0,4,2,3,5,1,3. Compute the estimate for the p derived in part 1

2. Relevant equations2. I know I have to start by finding a general form of ∏(ni+1)(1-p)ni p2 with ni where i goes from 1 to m.

3. The attempt at a solution3. So I get hung up on how to handle the (ni+1) term.

2. Relevant equations2. I know I have to start by finding a general form of ∏(ni+1)(1-p)ni p2 with ni where i goes from 1 to m.

There's a typo in your post. Witness that hanging .

I suspect you meant ##\Pi_{i=1}^m (n_i+1)(1-p)^{n_i} p^2##

3. The attempt at a solution3. So I get hung up on how to handle the (ni+1) term.

This isn't that hard. Let's start with m=1, i.e., one observation. Then that product is simply ##(n_1+1)(1-p)^{n_1}p^2##. With m=2, the product becomes ##(n_1+1)(1-p)^{n_1}p^2\,(n_2+1)(1-p)^{n_2}p^2##. Try collecting terms. You should get something of the form ##A (1-p)^B p^C##, where A, B, and C are related to the observed values. Now generalize to m observations.

I suspect you meant ##\Pi_{i=1}^m (n_i+1)(1-p)^{n_i} p^2##
This isn't that hard. Let's start with m=1, i.e., one observation. Then that product is simply ##(n_1+1)(1-p)^{n_1}p^2##. With m=2, the product becomes ##(n_1+1)(1-p)^{n_1}p^2\,(n_2+1)(1-p)^{n_2}p^2##. Try collecting terms. You should get something of the form ##A (1-p)^B p^C##, where A, B, and C are related to the observed values. Now generalize to m observations.

Where does this reach a maximum?

So I've collected terms for m=3 and get the following:

(n1+1)(n2+1)(n3+1)(1-p)n1(1-p)n2(1-p)n3p2p2p2

So by using your A(1-p)BpC:

I think B=∑ni
C=2n
A is where I'm still stuck. Am I missing something completely I can't seem to figure out how to reduced (n1+1)(n2+1)(n3+1)

And to find the maximum I need to set the derivative to 0 and find p the parameter the question ask for.

I think B=∑ni
C=2n
A is where I'm still stuck. Am I missing something completely I can't seem to figure out how to reduced (n1+1)(n2+1)(n3+1)

And to find the maximum I need to set the derivative to 0 and find p the parameter the question ask for.

Why would you care about reducing ##(n_1+1) (n_2+1) \cdots (n_m+1)##? It has no effect whatsoever on the optimal value of p. In other words, the p that maximizes f(p) is the same p that maximizes 1000*f(p) or (1/100)*f(p) or any K*f(p) for constant K.

Why would you care about reducing ##(n_1+1) (n_2+1) \cdots (n_m+1)##? It has no effect whatsoever on the optimal value of p. In other words, the p that maximizes f(p) is the same p that maximizes 1000*f(p) or (1/100)*f(p) or any K*f(p) for constant K.

Ok right I understand. I was confused.

So am I correct with the following next steps:

L(p)= ∏(ni+1)(1-p)∑nip2n

Then in the next take is to find l(p) which is l(p)=Ln(L(p))

So I end up with:

l(p)= Ln(∏(ni+1))+∑niLn(1-p)+2n*Ln(p)

Then taking the derivative of dl(p)/dp and setting that to zer
dl(p)/dp= 0+ [itex]\frac{∑n}{1-p}[/itex] +[itex]\frac{2n}{p}[/itex]